Optimal. Leaf size=26 \[ \frac {a x^2}{2}+\frac {b \text {ArcTan}\left (\sinh \left (c+d x^2\right )\right )}{2 d} \]
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Rubi [A]
time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {14, 5544, 3855}
\begin {gather*} \frac {a x^2}{2}+\frac {b \text {ArcTan}\left (\sinh \left (c+d x^2\right )\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 3855
Rule 5544
Rubi steps
\begin {align*} \int x \left (a+b \text {sech}\left (c+d x^2\right )\right ) \, dx &=\int \left (a x+b x \text {sech}\left (c+d x^2\right )\right ) \, dx\\ &=\frac {a x^2}{2}+b \int x \text {sech}\left (c+d x^2\right ) \, dx\\ &=\frac {a x^2}{2}+\frac {1}{2} b \text {Subst}\left (\int \text {sech}(c+d x) \, dx,x,x^2\right )\\ &=\frac {a x^2}{2}+\frac {b \tan ^{-1}\left (\sinh \left (c+d x^2\right )\right )}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 26, normalized size = 1.00 \begin {gather*} \frac {a x^2}{2}+\frac {b \text {ArcTan}\left (\sinh \left (c+d x^2\right )\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.86, size = 27, normalized size = 1.04
method | result | size |
derivativedivides | \(\frac {\left (d \,x^{2}+c \right ) a +b \arctan \left (\sinh \left (d \,x^{2}+c \right )\right )}{2 d}\) | \(27\) |
default | \(\frac {\left (d \,x^{2}+c \right ) a +b \arctan \left (\sinh \left (d \,x^{2}+c \right )\right )}{2 d}\) | \(27\) |
risch | \(\frac {a \,x^{2}}{2}+\frac {i b \ln \left ({\mathrm e}^{d \,x^{2}+c}+i\right )}{2 d}-\frac {i b \ln \left ({\mathrm e}^{d \,x^{2}+c}-i\right )}{2 d}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 22, normalized size = 0.85 \begin {gather*} \frac {1}{2} \, a x^{2} + \frac {b \arctan \left (\sinh \left (d x^{2} + c\right )\right )}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 33, normalized size = 1.27 \begin {gather*} \frac {a d x^{2} + 2 \, b \arctan \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right )\right )}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \left (a + b \operatorname {sech}{\left (c + d x^{2} \right )}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.38, size = 28, normalized size = 1.08 \begin {gather*} \frac {{\left (d x^{2} + c\right )} a}{2 \, d} + \frac {b \arctan \left (e^{\left (d x^{2} + c\right )}\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 42, normalized size = 1.62 \begin {gather*} \frac {a\,x^2}{2}+\frac {\mathrm {atan}\left (\frac {b\,{\mathrm {e}}^{d\,x^2}\,{\mathrm {e}}^c\,\sqrt {d^2}}{d\,\sqrt {b^2}}\right )\,\sqrt {b^2}}{\sqrt {d^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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